Angle Sums

  • Angle Sums

    Grade: 6th to 8th

    Examine the angles in a triangle, quadrilateral, pentagon, hexagon, heptagon or octagon. Can you find a relationship between the number of sides and the sum of the interior angles?



    • Choose a polygon, and reshape it by dragging the vertices to new locations. As the figure changes shape, the angle measures will automatically update.
    • Animation: For triangles and quadrilaterals, you can play an animated clip by clicking the image in the lower right corner. This movie will provide a visual proof for the value of the angle sum.


    Begin by exploring triangles and quadrilaterals. As you drag the vertices to reshape these polygons, occasionally play the animated clip in the lower right corner.
    • Is there a different result for different shapes?
    • Test your conjecture by creating a square, rhombus, parallelogram, and other quadrilaterals. For each type of shape, play the movie. How does the result change?
    Use the applet to examine pentagons, hexagons, heptagons and octagons, too. Do you notice a pattern? How does the sum of the angles change as the number of sides changes?
    • What happens to the angle sum as you reshape the polygon by moving the original vertices?
    • Find a formula that relates the number of sides (n) to the sum of the interior angle measures?

    Objectives and Standards

    NCTM Standards and Expectations
    • Geometry / Measurement
    • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
    • 6-8
    • Geometry